= Solution
There are about $\eta^{-1}\sim10^9$ <photons> per <baryon>. Even when the typical <photon> <energy> is below the <Hydrogen binding energy>, the high-energy tail can provide abundant ionizing <photons>. The nonrelativistic <Electron>'s translational <phase space> also favours the ionized state. Quantitatively, the <Saha equation> contains both suppressing factors $\eta$ and $(T/m_e)^{3/2}$. Recombination needs their tiny product to be overcome by $e^{I/T}$, so $I/T$ must be several tens, not of order one.
For an illustrative $\Omega_Bh^2=0.02$, the supplied baryon-to-photon relation gives $\eta=5.36\times10^{-10}$. With $m_e=5.11\times10^5\,\mathrm{eV}$, direct evaluation gives
$$
\begin{array}{c|ccc}
T\ (\mathrm{eV})&0.30&0.28&0.27\\
A(T)&45.1&1.04\times10^3&5.93\times10^3\\
X_e&0.138&0.0306&0.0129
\end{array}
$$
Thus \b[the equilibrium ionization fraction reaches a few per cent just below $0.3\,\mathrm{eV}$], while at $13.6\,\mathrm{eV}$ it is essentially unity. The exact <temperature> depends weakly on the <baryon> abundance and on the fraction used to define <cosmological recombination>. This is <small baryon abundance delays hydrogen recombination>. Recombination, <photon decoupling> and <residual electron freeze-out> are related but distinct: the last two involve rates and cannot be determined by <chemical equilibrium> alone.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-62-saha-ionization.png]
{title=Hydrogen-only Saha ionization fraction during cooling, with the three-per-cent crossing near 0.28 electronvolts for baryon-to-photon ratio 5.36 times ten to the minus ten}
Back to article page
